Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.

翻译：双宽度是最近由Bonnet、Kim、Thomassé和Watrigant引入的图宽度参数。针对两个图$G$和$H$以及某种图乘积$\star$，我们探讨以下问题：$G\star H$的双宽度是否受限于$G$与$H$的双宽度及其最大度数的函数？已知此类界限对强乘积成立（Bonnet、Geniet、Kim、Thomassé和Watrigant；SODA 2021）。我们证明，笛卡尔积、张量/直积、冠积、根积、替换积与锯齿积同样满足此类界限。对于字典序乘积，已知两个图乘积的双宽度恰好等于各图双宽度的最大值（Bonnet、Kim、Reinald、Thomassé和Watrigant；IPEC 2021）。相反，对于模积，我们证明无法成立任何界限。此外，我们通过实例表明许多界限是紧的，并针对特定图类给出了改进后的界限。